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<h1 class="ltx_title ltx_title_section">Averaged shifted histograms</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">This visualization algorithm uses a density estimation approach based
on averaged shifted histograms <cite class="ltx_cite">[<a href="#bib.bib30" title="Averaged shifted histograms: effective nonparametric density estimators in several dimensions" class="ltx_ref">1</a>]</cite>. In the one-dimensional
case, this method works by averaging <img id="Sx1.p1.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi4.png" width="15" height="12" alt="n"> histograms with the same
bin width <img id="Sx1.p1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi7.png" width="17" height="12" alt="w">, but with the origin of each histogram shifted by <img id="Sx1.p1.m3" class="ltx_Math" style="vertical-align:-8px" src="mi/mi1.png" width="19" height="22" alt="\frac{w}{n}">
from the previous histogram. In the multidimensional case, there are
<img id="Sx1.p1.m4" class="ltx_Math" style="vertical-align:-2px" src="mi/mi5.png" width="23" height="19" alt="n^{d}"> multidimensional histograms averaged in total, i.e., for
<img id="Sx1.p1.m5" class="ltx_Math" style="vertical-align:-2px" src="mi/mi4.png" width="15" height="12" alt="n"> shifts in each of the <img id="Sx1.p1.m6" class="ltx_Math" style="vertical-align:-2px" src="mi/mi3.png" width="13" height="16" alt="d"> dimensions. In our implementation,
the width of the histogram bin is determined as <img id="Sx1.p1.m7" class="ltx_Math" style="vertical-align:-2px" src="mi/mi6.png" width="61" height="12" alt="w=na">, where <img id="Sx1.p1.m8" class="ltx_Math" style="vertical-align:-2px" src="mi/mi2.png" width="13" height="12" alt="a">
is the pixel size of the super-resolution image. The number of shifts
<img id="Sx1.p1.m9" class="ltx_Math" style="vertical-align:-2px" src="mi/mi4.png" width="15" height="12" alt="n"> in the lateral and axial directions can be specified independently.</p>
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<h2 class="ltx_title ltx_title_subsection">See also</h2>

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<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="Rendering.html" title="" class="ltx_ref">Visualization methods</a></p>
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<h1 class="ltx_title ltx_title_bibliography">References</h1>

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<li id="bib.bib30" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">D. W. Scott</span><span class="ltx_text ltx_bib_year">(1985)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Averaged shifted histograms: effective nonparametric density estimators in several dimensions</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">The Annals of Statistics</span> <span class="ltx_text ltx_bib_volume">13</span> (<span class="ltx_text ltx_bib_number">3</span>), <span class="ltx_text ltx_bib_pages"> pp. pp. 1024–1040</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://www.jstor.org/stable/2241123" title="" class="ltx_ref ltx_bib_external">Link</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.p1" title="Averaged shifted histograms" class="ltx_ref"><span class="ltx_text ltx_ref_title">Averaged shifted histograms</span></a>.
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